The ellipticities of Galactic and LMC globular clusters

The globular clusters of the LMC are found to be significantly more elliptical than Galactic globular clusters, but very similar in virtually all other respects. The ellipticity of the LMC globular clusters is shown not be correlated with the age or mass of those clusters. It is proposed that the ellipticity differences are caused by the different strengths of the tidal fields in the LMC and the Galaxy. The strong Galactic tidal field erases initial velocity anisotropies and removes angular momentum from globular clusters making them more spherical. The tidal field of the LMC is not strong enough to perform these tasks and its globular clusters remain close to their initial states.Comment: 3 pages LaTeX file with 3 figures incorporated accepted for publication in MNRAS. Also available by e-mailing spg, or by ftp from ftp://star-www.maps.susx.ac.uk/pub/papers/spg/ellip.ps.

On commuting varieties of nilradicals of Borel subalgebras of reductive Lie algebras

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field \mathbbm k of characteristic zero. We consider the commuting variety $\mathcal C(\mathfrak u)$ of the nilradical $\mathfrak u$ of the Lie algebra $\mathfrak b$ of a Borel subgroup $B$ of $G$. In case $B$ acts on $\mathfrak u$ with only a finite number of orbits, we verify that $\mathcal C(\mathfrak u)$ is equidimensional and that the irreducible components are in correspondence with the {\em distinguished} $B$-orbits in $\mathfrak u$. We observe that in general $\mathcal C(\mathfrak u)$ is not equidimensional, and determine the irreducible components of $\mathcal C(\mathfrak u)$ in the minimal cases where there are infinitely many $B$-orbits in $\mathfrak u$.Comment: 10 page

On commuting varieties of parabolic subalgebras

Let $G$ be a connected reductive algebraic group over an algebraically closed field $k$, and assume that the characteristic of $k$ is zero or a pretty good prime for $G$. Let $P$ be a parabolic subgroup of $G$ and let $\mathfrak p$ be the Lie algebra of $P$. We consider the commuting variety $\mathcal C(\mathfrak p) = \{(X,Y) \in \mathfrak p \times \mathfrak p \mid [X,Y] = 0\}$. Our main theorem gives a necessary and sufficient condition for irreducibility of $\mathcal C(\mathfrak p)$ in terms of the modality of the adjoint action of $P$ on the nilpotent variety of $\mathfrak p$. As a consequence, for the case $P = B$ a Borel subgroup of $G$, we give a classification of when $\mathcal C(\mathfrak b)$ is irreducible; this builds on a partial classification given by Keeton. Further, in cases where $\mathcal C(\mathfrak p)$ is irreducible, we consider whether $\mathcal C(\mathfrak p)$ is a normal variety. In particular, this leads to a classification of when $\mathcal C(\mathfrak b)$ is normal.Comment: 19 pages; minor update